To find the exact value of the length of the square, we can follow these steps:

Step 1: Recall the formula for the perimeter of a square.

The perimeter $P$ of a square is given by the formula: $P = 4 \times \text{side length}$ Let the side length of the square be $s$.

Step 2: Set up the equation using the perimeter.

We are told that the perimeter is 75 and that the side length is expressed as $2x + 5$. So we can write the equation: $4 \times (2x + 5) = 75$

Step 3: Solve for $x$.

First, simplify the equation: $4(2x + 5) = 75$ Distribute the 4: $8x + 20 = 75$ Subtract 20 from both sides: $8x = 75 - 20$ $8x = 55$ Now, divide both sides by 8: $x = \frac{55}{8}$

Step 4: Find the exact side length.

Now that we know $x = \frac{55}{8}$, substitute this value into the expression for the side length: $\text{side length} = 2x + 5 = 2\left(\frac{55}{8}\right) + 5$ Simplify: $\text{side length} = \frac{110}{8} + 5 = \frac{110}{8} + \frac{40}{8} = \frac{150}{8} = \frac{75}{4}$

Final Answer:

The exact length of the side of the square is $\frac{75}{4}$ or 18.75 units.

Would you like more details on any part of this solution?

Here are some related questions:

1. What happens to the side length if the perimeter of the square is doubled?
2. How would the equation change if the perimeter were given as a different expression involving $x$?
3. Can you use this method to solve for the area of the square if you were given the perimeter?
4. What is the relationship between the side length and the diagonal of a square?
5. How can this method be applied to solve for the side length of a rectangle with a known perimeter?

Tip: When dealing with algebraic expressions for geometric shapes, always start by writing down the formula relating to the shape (like perimeter, area, etc.) and then substitute the given values!